Elliptic singularities on log symplectic manifolds and Feigin--Odesskii Poisson brackets

Abstract

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities ${\tilde{E}}_6,{\tilde{E}}_7$ and ${\tilde{E}}_8$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii's Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic.

33 pages, comments welcome

References in corpus (3)

• Stable generalized complex structures
• Geometric syzygies of elliptic normal curves and their secant varieties
• A characterization of diagonal Poisson structures

Cited by in corpus (8)

• Constructions and classifications of projective Poisson varieties
• Reduction theory for singular symplectic manifolds and singular forms on moduli spaces
• The tropical momentum map: a classification of toric log symplectic manifolds
• Mixed Hodge structures in log symplectic geometry
• A local Torelli theorem for log symplectic manifolds
• A Bogomolov unobstructedness theorem for log-symplectic manifolds in general position
• An Example of Rank 2 Poisson Structure which is Stable Under Deformations
• Elliptic log symplectic brackets on projective bundles